C. Review of Thermodynamics and Statistical Mechanics

specifies the change in the internal energy arising from small independent changes in the entropy , the volume , and the number of particles . Equation (C.1) shows that the internal energy is a function of these three variables, , and that the temperature , the pressure , and the FERMI energy (also called the chemical potential) are related to the partial derivatives of

In practice, however, experiments are usually performed at fixed and it is convenient to make LEGENDRE transformation to the variables or . The resulting functions are known as

with the corresponding differential and coefficients

Although , , , and represent equivalent ways of describing the same system, their natural independent variables differ in one important way. In particular, the set consists entirely of extensive variables, proportional to the actual amount of matter present. The transformation to and then to or may be interpreted as reducing the number of extensive variables in favor of intensive ones that are independent of the total amount of matter.

To this point, only macroscopic thermodynamics has been discussed. The
microscopic content of the theory must be added separately through statistical
mechanics, which relates the thermodynamic functions to the HAMILTONian of
the many-particle system. The elementary discussions of statistical mechanics
usually consider systems containing a fixed number of particles. This approach,
which is refereed to as *canonical ensemble*, is too restricted for our
purposes. To include the possibility of a variable number of particles the
*grand canonical ensemble* can be employed. For a grand canonical
ensemble at FERMI energy
and temperature , the grand partition
function is defined as

where denotes the set of all states for a fixed number of particles , and the sum implied in the trace is over both and . Short-hand notation has been introduced, where is the BOLTZMANN constant. A fundamental result from statistical mechanics states that

which allows one to compute all the macroscopic equilibrium thermodynamics from the grand partition function. The statistical operator corresponding to (C.6) is given by

For any operator , the ensemble average is achieved with the prescription

By applying these results the properties of a gas of non-interacting Bosons or FERMIons can be studied. If (C.6) is written out in detail with the complete set of states in the abstract occupation number HILBERT space, one gets

(C.10) |

Since these states are eigen-states of the non-interacting HAMILTONian and the number operator , both operators can be replaced by their eigen-values

(C.11) |

The exponential is now a number and is equivalent to a product of exponentials. Therefore, the sum over expectation values factor into a product of traces